Sequences and series (Studies)

Question 1

What is an arithmetic sequence?

Answer 1

A list of numbers that change by adding or subtracting the same amount each time.

Question 2

What is a “common difference, d”?

Answer 2

It’s the amount of increase (or decrease when negative) between terms of an arithmetic sequence.

Question 3

How can you find d when given two consecutive terms (like u₃ and u₄) of an arithmetic series?

Answer 3

You subtract the terms. Always the later term minus the earlier one.

u₄ - u₃ = d

Question 4

How can you find d when given two non-consecutive terms (like u₄ and u₈) of an arithmetic series?

Answer 4

You set up a system of equations and solve it (with linSolve).

1. Write an equation for each u_n you know.
2. Plug these equations into linSolve.

Question 5

How can you find u_n if you already know u₁ and d of an arithmetic series?

Answer 5

You plug it into the formula from your booklet.

Question 6

What’s the difference between u_n and n?

Answer 6

• n* tells you the placement of the term. Like, 1st, 2nd, 37th, etc.
• u_n* is the actual value of the term.

Question 7

What’s the difference between u_n and S_n?

Answer 7

• u_n* is the value of only the nth term
• S_n* is what you get if you add together all of the n terms.

Question 8

What are context clues that you should be finding S_n?

Answer 8

You see these words in the question:

• calculate/find the total…
• the sum of the first n terms…

Question 9

Your formula booklet has two equations for S_n of an arithmetic sequence.

Which one should you use?

Answer 9

It all comes down to whether you know u_n or d.

• Use the first one if you only know u₁ and d.
• Use the second one if you know both the 1st and last terms (u₁ and u_n).

Question 10

What is a geometric sequence?

Answer 10

A list of numbers that change by multiplying (or dividing) the same amount each time.

Question 11

What is a “common ratio, r”?

Answer 11

It’s the multiple of increase (or decrease when a fraction less than 1) between terms of a geometric sequence.

Question 12

How can you find d when given two consecutive terms (like u₃ and u₄) of an arithmetic series?

Answer 12

You divide the terms. Always the later term divided by the earlier one.

r = u₄ ÷ u₃

Question 13

How can you find r when given two non-consecutive terms (like u₄ and u₈) of an geometric series?

Answer 13

You set up a system of equations and solve it (with nSolve).

1. Write an equation for each u_n you know.
2. Solve one equation for u₁ by hand. (It will still have r in it).
3. Substitute this equation into the other. (Now there are only rs.)
4. Solve this equation with nSolve.

Question 14

How can you find u_n if you already know u₁ and r of a geometric series?

Answer 14

You plug it into the formula from your booklet.

Question 15

Your formula booklet has two equations for Sn of a geometric sequence.

Which one should you use?

Answer 15

It literally makes no difference. Some people argue that one is better than the other, depending on whether r is greater or less than 1. But this is nonsense. Use either one; it doesn’t matter at all.

Question 16

What type of sequence (arithmetic or geometric) is it if you are asked about simple interest?

Answer 16

Arithmetic!

Question 17

If the question talks about “percent increase” or “percent decrease”, what type of sequence is it?

Answer 17

Geometric!

r = 1 + “percent increase”

r = 1 - “percent decrease”

Question 18

How can you tell when a series exceeds some value?

Answer 18

You set up an equation for when S_n equals that value.

Nsolve to find n.

If n is a decimal number, check S_n for the integer values above and below the decimal number.